Operator-valued Kernels and Control of Infinite dimensional Dynamic Systems

TL;DR

This paper extends the connection between the Linear Quadratic Regulator (LQR) and kernel methods from finite to infinite dimensional systems, enabling new approaches to control PDEs using kernel-based techniques.

Contribution

It generalizes the kernel-LQR relationship to infinite dimensional systems, providing a formula based on the Riccati equation for controlling PDEs.

Findings

Derived a kernel-based formula for infinite dimensional LQR control

Extended the finite-dimensional kernel control framework to PDEs

Facilitated the use of representer theorems in infinite dimensional control

Abstract

The Linear Quadratic Regulator (LQR), which is arguably the most classical problem in control theory, was recently related to kernel methods in (Aubin-Frankowski, SICON, 2021) for finite dimensional systems. We show that this result extends to infinite dimensional systems, i.e.\ control of linear partial differential equations. The quadratic objective paired with the linear dynamics encode the relevant kernel, defining a Hilbert space of controlled trajectories, for which we obtain a concise formula based on the solution of the differential Riccati equation. This paves the way to applying representer theorems from kernel methods to solve infinite dimensional optimal control problems.